Logs and bones

John Napier, a Scottish mathematician, took it on himself to make arithmetic easier for the masses. He fancied himself as an inventor and set about the task in this spirit.

John Napier (1550-1617)

Napier is best known for the idea of using logarithms in 1614 to make arithmetic easier - but he didn't invent the closely related slide rule. Every number has a logarithm and adding logarithms is the same a multiplying the original values. If you have two numbers that you want to multiply you look up their logs, add the logs together and then look up the antilog. Of course dividing is just a matter of subtraction.

Logs make multiplication and division easy - but only if you have suitable tables in which to look up the logs and anti-logs. It was the need to compute such tables that started Charles Babbage off thinking about his difference machine and eventually to formulate his conception of the modern computer.

As well as logarithms, Napier's name is forever coupled with Bones. Napier's Bones (1617) were a set of rods with multiplication tables written out on them. Users could select rods for a particular multiplication and then, by adding up the partial products, get a final answer. As well as coping with multiplication a rod was included for square and cube roots.

Napier's Bones

Napier's Bones were based on a method of multiplication known much earlier by the name Gelosia. You write the two products along the sides of a grid and write out the results in the body of the grid. As each pair of digits can produce a two digit number any "tens" part is written in the top diagonal and the "units" part in the bottom. Finally adding along the diagonals gives the final answer:

Napier's Bones were multiplication tables written out in the Gelosia form. You selected the bones corresponding to the digits of the first value and read of the partial products corresponding to the second value - adding these up along the diagonals gives the final result:

The fact that the Bones were important is more a measure of how poor the population were at arithmetic rather than how clever or sophisticated the Bones were. One thing they were very certainly not was the forerunner of the slide rule.

The slide rule seems an obvious enough invention once you know about logarithms but Napier didn't think of it - William Oughtred did some years later. Oughtred was a vicar and amateur mathematician who also invented the use of the x sign for multiplication.

William Oughtred (1574-1660)

The slide rule is a very simple analog computer. It has scales marked up so that lengths are proportional to logs. By moving the two parts of the rule lengths can be added and the result of the multiplication read off an anti-log scale. Using a slide rule became a skill that every engineer and scientist acquired until remarkable recently.